On commutative finite-dimensional nilalgebras
type: Item
instance of: scholarly article
title: On commutative finite-dimensional nilalgebras
zbMATH Open document ID: 1357.17001
author: Claudia I. Garcia
author: Juan C. Gutierrez Fernandez
published in: São Paulo Journal of Mathematical Sciences
publication date: 2016-11-08T00:00:00Z
review text: The authors survey certain results related to Albert's question: is every finite-dimensional commutative power-associative nilalgebra over a field solvable? In particular, the question has been answered in the affirmative for algebras of dimension \(\leq8\) provided that the characteristic of the ground field is different from 2, 3, and 5. In addition, the paper collects elementary proofs (mainly due to \textit{B. Mathes} et al. [Linear Algebra Appl. 149, 215--225 (1991; Zbl 0722.15016)]) for some results by \textit{M. Gerstenhaber} [Am. J. Math. 80, 614--622 (1958; Zbl 0085.26204)] and other authors about linear spaces of nilpotent matrices. These results are relevant in the context of Albert's question since for \(a\in A\), where \(A\) is a commutative power-associative nilalgebra of dimension \(n\) and nilindex \(t\), the left multiplication operator \(L_a: A\to A\) defined by \(L_a(x):=ax\) is known to be nilpotent of index \(\leq 2t-3\) [\textit{M. Gerstenhaber}, Duke Math. J. 27, 21--31 (1960; Zbl 0213.04101)]. Hence the linear space \(\{L_a\mid a\in A\}\) is isomorphic to a linear space of nilpotent \(n\times n\)-matrices.
Mathematics Subject Classification ID: 17A05
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 6649138
zbMATH Keywords: power-associative algebra
zbMATH Keywords: nilalgebra
zbMATH Keywords: solvable algebra
zbMATH Keywords: nilpotent matrix
zbMATH Keywords: Albert's question
MaRDI profile type: Publication
full work available at URL: s40863-015-0035-z
OpenAlex ID: W2285642462
Wikidata QID: Q108437987
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DOI: 10.1007/S40863-015-0035-Z
description: scientific article; zbMATH DE number 6649138